#Bespiel zur verdeckten / Scheinkorrelation

zeit <- 1:10
x <- c(38,37,37,37,35,34,34,34,33,31)
y <- c(49,48,47,46,46,46,44,44,43,40)

#a) 
(xbar <- mean(x))
(ybar <- mean(y))

(sxy <- sum((x-xbar)*(y-ybar)))
(sx2 <- sum((x-xbar)^2))
(sy2 <- sum((y-ybar)^2))

sxy/sqrt(sx2*sy2)
rxy <- round(cor(x, y), 2)

#b) 
tbar <- mean(zeit)
(st2 <- sum((zeit-tbar)^2))
(stx <- sum((zeit-tbar)*(x-xbar))) 
(sty <- sum((zeit-tbar)*(y-ybar)))


(bx <- stx/st2)
(ax <- xbar - bx*tbar)

(by <- sty/st2)
(ay <- ybar - by*tbar)

lm(x~zeit)
residuals(lm(x~zeit))
lm(y~zeit)
residuals(lm(y~zeit))

cor(residuals(lm(x~zeit)),residuals(lm(y~zeit)))

# Alternativ: partieller Korrelationskoeffizient:

stx/sqrt(st2*sx2)
rxt <- round(cor(x, zeit),2)

sty/sqrt(st2*sy2)
ryt <- round(cor(y, zeit),2)

(cor_b <- (rxy - rxt*ryt)/sqrt((1-rxt^2)*(1-ryt^2)))
(cor(x,y) - cor(x,zeit)*cor(y,zeit))/sqrt((1-cor(x,zeit)^2)*(1-cor(y,zeit)^2))